WBJEE · Maths · Ellipse

The locus of the mid-points of the chords of an ellipse \(x^{2}+4 y^{2}=4\) that are drawn from the positive end of the minor axis, is

A a circle with centre \(\left(\frac{1}{2}, 0\right)\) and radius 1
B a parabola with focus \(\left(\frac{1}{2}, 0\right)\) and directrix \(x=-1\)
C an ellipse with centre \(\left(0, \frac{1}{2}\right),\) major axis 1 and minor axis \(\frac{1}{2}\)
D a hyperbola with centre \(\left(0, \frac{1}{2}\right),\) transverse axis 1 and conjugate axis \(\frac{1}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(C) an ellipse with centre \(\left(0, \frac{1}{2}\right),\) major axis 1 and minor axis \(\frac{1}{2}\)

Step-by-step Solution

Detailed explanation

Given equation of an ellipse is \[ x^{2}+4 y^{2}=4 \] \(\Rightarrow \quad \frac{x^{2}}{4}+\frac{y^{2}}{1}=1\) \(\therefore\) Coordinate of positive end of minor axis is \(B(0,1)\) Let mid-point of the chord \(B P\) is \(M(h, k)\) Then,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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